Problem: 23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once?
Answer: Note that if each person shakes hands with every other person, then the number of handshakes is maximized. There are $\binom{23}{2} = \frac{(23)(22)}{2} = (23)(11) = 230+23 = \boxed{253}$ ways to choose two people to form a handshake.